Question 81
Let $M$ and $N$ be the midpoints of the sides $AC$ and $BC$ of $\triangle ABC$ ($AC \gt BC$) and let the bisector of $\angle B$ intersect the segment $MN$ at a point $P$. The incircle of $\triangle ABC$ has center $I$ and is tangent to $BC$ at a point $Q$. Denote by $R$ the intersection point of the perpendiculars from $P$ and $Q$ to $MN$ and $BC$, respectively, and by $S$ the intersection point of the lines $AB$ and $RN$.
a) Prove that the quadrilateral *PCQI* is cyclic.
b) Express the length of the segment *BS* by the lengths *a*, *b*, *c* of the sides of $\triangle ABC$.
a) Prove that the quadrilateral *PCQI* is cyclic.
b) Express the length of the segment *BS* by the lengths *a*, *b*, *c* of the sides of $\triangle ABC$.